 Teichmuller lift properties

For residue field ${\mathbb{F}}_q$, these two properties characterize the Teichmuller lift on every residue class, including zero. The Hensel lemma applies to $T^q-T$, whose derivative reduces to $-1$. Uniqueness implies $[ab]=[a][b]$, $[0]=0$, $[1]=1$, and $[a^p]=[a{]}^p$. In mixed characteristic this multiplicative section need not be additive.